Resonances, waves and fields: Transforming electromagnetic fields and Maxwell's equations

Tuesday, July 12, 2011

12. Transforming electromagnetic fields and Maxwell's equations

Relativity has its roots in a problem associated with Maxwell's electromagnetic theory: that there was no obvious way to address the question of reference frames in Maxwell's equations. It was further prompted by the Michelson-Morley experiment which indicated that the speed of light as measured is independent of the velocity of the reference frame from which it is measured.

"Relativity" was created to make a world in which all objects, when moving at very great velocities, would behave in a manner consistent with their internal structures being determined by Maxwell's equations and the constant speed of light. Part of this world was a set of "Lorentz" transforms which allowed us to convert observations taken in one reference frame into observations in another reference frame.

Lorentz transforms have the property that they preserve the form of the operators in both Maxwell's equations and the electromagnetic wave equations. At the same time, this does not mean that the electromagetic variables, the electric and magnetic fields, and the charge and current densities, stay the same when changing reference frames. In fact, we shall find, in the next few chapters, that all these variables do change.

We shall find, that when we change reference frames a pure electric field becomes a mix of electric and magnetic fields. The same is true of a pure magnetic field. Also a pure charge density becomes a mix of charge density and current density. A pure current density becomes a mix of current and charge density.

In a sense, it is obvious. After all in a reference frame where a charge is stationary, it is a pure charge (not a current) and it creates a pure electric field. However, when we change reference frames so that the charge appears to be moving, we have a moving charge which is not only a charge, but also is a moving charge, i.e. an electrical current. In this case we would expect not only an electric field (from the charge) but also a magnetic field (from the electrical current). The pure electric field is changed into a mix of electric field plus magnetic field with the addition of motion.

We shall derive the equations for these transformations. In the final four chapters we shall come full circle and transform not only the operator part of Maxwell's equations, but also the electric and magnetic fields and also the current and charge densities. With quite a lot of brute force work, we shall show that the set of equations in fact do transform into an identical looking set of Maxwell's equations in the new reference frame.